For the central black hole, we assume a Schwarzschild black hole with mass \(M_{*}\)  = 10\(M_{\odot }\) .
First, we set model parameters of a specific internal energy \(\mathcal {E}\) 
and a specific angular momentum \(\lambda \) 
which are responsible for the standing shock.
Then, given an outer radial boundary \(R_{\rm out}\)  and an input accretion rate
\(\dot{M}_{\rm input}\) , we obtain flow variables of the density \(\rho _{\rm out}\) , the radial velocity \(v_{\rm out}\) 
and the sound velocity \(a_{\rm out}\)  at the outer radial boundary from 1.5D transonic solutions [1]},
under the vertical hydrostatic equilibrium assumption of the flow. For the given sound velocity \(a_{\rm out}\) , the gas temperature
\(T_{\rm out}\)  is obtained from \(a_{\rm out}^{2} ={\gamma R_{\rm G}T_{\rm out}} /\mu \)  and \(\gamma aT_{\rm out}^{4}/ {3{\rho }_{\rm out}}\) 
for gas-pressure dominant and radiation-pressure dominant cases, respectively, where \(\gamma \)  is the specific heat ratio.
Accordingly, if the input gas is gas-pressure dominant,
the temperature \(T_{\rm out}\)  is determined independent of input density \(\rho _{\rm out}\) . On the other hand, if the radiation pressure is dominant, the input gas temperature \(T_{\rm out}\)  and the radiation energy density \((E_{0})_{\rm out} (= a T_{\rm out}^{4})\) 
increases with increasing input density, assuming the local thermal equilibrium at the outer boundary.
In 1.5D transonic equations, we have the specific energy \(\mathcal {E}\) 
, that is, Bernoulli constant, given by
\(\mathcal {E}= \frac{{v}_{\rm out}^{2}}{2} + \frac{a_{\rm out}^{2}}{\gamma - 1} + \frac{\lambda ^{2}}{2r_{\rm out}^{2}} -\frac{1}{2(r_{\rm out}-1)},\) 
